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  • A B1A003 Total No. of pages:2

    Page 1 of 2

    Reg. No._______________ Name:__________________________

    APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

    THIRD SEMESTER B.TECH DEGREE EXAMINATION, DEC 2016

    Course Code: MA201

    Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

    Max. Marks: 100 Duration:3. Hours

    PART A

    (Answer any two questions)

    1.a Show that � = �� − 3���is harmonic and hence find its harmonic conjugate. (8)

    b Find the image of �� − �

    � � ≤

    � under the transformation =

    � . Also find the fixed points

    of the transformation � = �

    � (7)

    2.a Define an analytic function and prove that an analytic function of constant modulus is

    constant. (8)

    b Find the linear fractional transformation that maps �� = 0, �� = 1, �� = ∞onto

    �� = −1, �� = −�, �� = 1 respectively. (7)

    3.a Show that �(�) = ������� − �������� is differentiable everywhere. Find

    its derivative. (8)

    b Find the image of the lines � = � and � = �, where �&�are constants, under the

    transformation � = ����. (7)

    PART B

    (Answer any two questions)

    4.a Evaluate ∫ �� (�) �� �

    where � is a straight line from 0 to 1 + 2�. (7)

    b Show that ∫ ��

    ���� =

    �√�

    � (8)

    5.a Integrate ��

    ���� counterclockwise around the circle |� − 1 − �| =

    � by Cauchy’s

    Integral Formula. (7)

    b Evaluate ∫ ����

    ������� ��

    � where � is |� − 2 − �| = 3.5 by Cauchy’s Residue Theorem

    (8)

    6.a If �(�) = �

    �� find the Taylor series that converges in |� − �| < �and the Laurent’s

    series that converges in |� − �| > �. (8)

    b Define three types of isolated singularities with an example for each. (7)

    Department of Mechanical Engineering, SCMS School of Engineering and Technology.

  • A B1A003 Total No. of pages:2

    Page 2 of 2

    PART C

    (Answer any two questions)

    7.a Solve by Gauss Elimination:

    �� − �� + �� = 0,

    −�� + �� − �� = 0,

    10 �� + 25 �� = 90,

    20 �� + 10 �� = 80. (5)

    b Find the rank. Also find a basis for the row space and column space for

    � 0 1 0 −1 0 −4 0 4 0

    � (5)

    c Find out what type of conic section the quadratic form

    � = 17 �� − 30 �� + 17 �� = 128 represents and transform it to the principal

    axes. (10)

    8.a Find whether the vectors [1 2−1 3], [2 −13 2]��� [−1 8−9 5] are

    linearly dependent. (5)

    b Show that the matrix � = � 1 2 2 −2

    � is symmetric. Find the spectrum. (5)

    c Diagonalise � = � 8 −6 2 −6 7 −4 2 −4 3

    � (10)

    9. a. Determine whether the matrix

    ⎣ ⎢ ⎢ ⎡ 1 0 0

    0 1 √2

    � −1 √2

    0 1 √2

    � 1 √2

    � ⎦ ⎥ ⎥ ⎤ is orthogonal? (5)

    b. Find the Eigen values and Eigen vectors of � 1 1 2 −1 2 1 0 1 3

    � (5)

    c. Define a Vector Space with an example. (10)

    Department of Mechanical Engineering, SCMS School of Engineering and Technology.

  • A B3A005 Pages:2

    Page 1 of 2

    Reg. No._____________ Name:_____________________

    APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

    THIRD SEMESTER B.TECH DEGREE EXAMINATION, MARCH 2017

    MA 201: LINEAR ALGEBRA AND COMPLEX ANALYSIS

    Max. Marks: 100 Duration: 3 Hours

    PART A

    Answer any 2 questions

    1. a. Check whether the following functions are analytic or not. Justify your answer.

    i)   zzf z (4)

    ii)   2zzf 

    (4)

    b. Show that   zzf sin is analytic for all z. Find  zf  (7)

    2. a. Show that 323 yyxv  is harmonic and find the corresponding analytic function

         yxivyxuzf ,,  (8)

    b. Find the image of 10  x , 1 2

    1  y under the mapping zew  (7)

    3. a. Find the linear fractional transformation that carries �� = −2, �� = 0 and �� = 2

    on to the points �� = ∞, �� = 1

    4� and �� = 3

    8� . Hence find the image of x-axis.(7)

    b. Find the image of the rectangular region   x , bya  under the mapping

    zw sin (8)

    PART B

    Answer any 2 questions

    4. a. Evaluate ∫ |�|�� �

    where

    i) C is the line segment joining -i and i (3)

    ii) C is the unit circle in the left of half plane (4)

    b. Verify Cauchy’s integral theorem for �� taken over the boundary of the rectangle

    with vertices -1, 1, 1+i, -1+i in the counter clockwise sense. (8)

    5. a. Find the Laurent’s series expansion of   21

    1

    z zf

      which is convergent in

    i) |� − 1| < 2 (4)

    ii) |� − 1| > 2 (4)

    b. Determine the nature and type of singularities of

    i) 2

    2

    z

    e z (3)

    Department of Mechanical Engineering, SCMS School of Engineering and Technology.

  • A B3A005 Pages:2

    Page 2 of 2

    ii) � sin (� � )

    (4)

    6. a. Use residue theorem to evaluate    

    dz zz

    zz

    C

      

    1312

    52330 2

    2

    where C is 1z (7)

    b. Evaluate  

    dx x

     

    0 221

    1 using residue theorem. (8)

    PART C

    Answer any 2 questions

    7. a. Solve the following by Gauss elimination

    y + z – 2w = 0, 2x – 3y – 3z + 6w = 2, 4x + y + z – 2w = 4 (6)

    b. Reduce to Echelon form and hence find the rank of the matrix

      

      

    

    1502121

    5424426

    2203

    (6)

    c. Find a basis for the null space of

      

      

     

    402

    840

    022

    (8)

    8. a. i) Are the vectors (3 -1 4), (6 7 5) and (9 6 9) linearly dependent or

    independent? Justify your answer. (5)

    ii) Is all vectors  zyx ,, in ℝ� with 04  zxy form a vector space over the field

    of real numbers? Give reasons for your answer. (5)

    b. i) Find a matrix C such that xCxTQ  where

    2 331

    2 221

    2 1 5243 xxxxxxxQ  (4)

    ii) Obtain the matrix of transformation

    y1 = cos θ x1 – sin θ x2, y2 = sin θ x1 + cos θ x2

    Prove that it is orthogonal. Obtain the inverse transformation. (6)

    9. a. Find the eigenvalues, eigenvectors and bases and dimensions for each Eigen space

    of

      

      

    

    

    021

    612

    322

    A

    (10)

    b. Find out what type of conic section, the quadratic form 128173017 2221 2 1  xxxx

    and transform it to principal axes. (10)

    Department of Mechanical Engineering, SCMS School of Engineering and Technology.